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Chapter 5: Problem 136

Write each number in scientific notation. $$ 0.0072 $$

Short Answer

Expert verified
7.2 \times 10^{-3}

Step by step solution

01

Identify the significant digits

The significant digits in the number 0.0072 are 7 and 2.
02

Form the coefficient

Place the decimal point after the first significant digit to form the coefficient. For 0.0072, the coefficient is 7.2.
03

Determine the exponent

Count the number of places the decimal point has moved to the right to form the coefficient. It moves 3 places, so the exponent is -3.
04

Write in scientific notation

Combine the coefficient and the exponent to express the number in scientific notation. Therefore, 0.0072 in scientific notation is written as \[ 7.2 \times 10^{-3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Digits
When working with scientific notation, the first step is to identify the significant digits in the number. Significant digits are the meaningful digits in a number, which contribute to its precision. In the given example, 0.0072, the significant digits are 7 and 2.
These digits do not include leading zeros, as they merely act as placeholders. Significant digits help us in forming the coefficient for scientific notation.
Coefficient
After identifying the significant digits, the next step is to form the coefficient. The coefficient is a number typically between 1 and 10, created by placing the decimal point immediately after the first significant digit.
For the number 0.0072, the first significant digit is 7, so placing the decimal point after 7 gives us 7.2. This number is now our coefficient. The goal of forming the coefficient is to simplify the original number while maintaining its value.
Exponent
Once the coefficient is formed, the exponent must be determined. The exponent shows how many places the decimal point needs to be moved to convert the original number into the coefficient.
In the example 0.0072, the decimal point is moved 3 places to the right to change it to 7.2. Therefore, the exponent is -3. This indicates that the coefficient (7.2) must be divided by 10 raised to the power of 3 to revert back to the original number.
Decimal Point
The decimal point plays a crucial role in both the original number and the formation of the coefficient. It's the key element that gets repositioned to convert the number into scientific notation.
In the case of 0.0072, the decimal point starts after the first three zeros. By shifting it three places to the right, it helps us transition from 0.0072 to 7.2. Understanding the decimal point's movement is imperative to correctly identifying the exponent and forming the scientific notation.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{p^{2} q^{-1}}{2 p^{-2}}\right)^{2} \cdot\left(\frac{p^{3} \cdot 4 q^{-2}}{3 q^{-5}}\right)^{-1} \cdot\left(\frac{p q^{-5}}{q^{-2}}\right)^{3} $$

Use the FOIL method to find each product. $$ (6 c-d)(2 c+3 d) $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ 7 k^{2}(-2 k)\left(4 k^{-5}\right)^{0} $$

Evaluate. Express answers in standard notation. $$ \frac{5 \times 10^{-3}}{25 \times 10^{2}} $$

$$\text {Let } f(x)=x^{2}-9, g(x)=2 x, \text { and } h(x)=x-3 . \text { Find each of the following.}$$ $$(f g)(-2)$$
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